Non-symmetric elliptic operators on bounded Lipschitz
domains in the plane

David J. Rule
Abstract:
We consider divergence form elliptic operators
in
with a coefficient
matrix
of bounded measurable functions independent of the
-direction.
The aim of this note is to demonstrate how the
proof of the main theorem in [4] can be modified to bounded
Lipschitz domains. The original theorem states that the
Neumann
and regularity problems are solvable for
for some
in domains of the form
, where
is a Lipschitz function. The exponent
depends only on the
ellipticity constants and the Lipschitz constant of
.
The principal modification of the argument for the original
result is to prove the boundedness of the layer potentials on domains
of the form
,
for a fixed unit vector
and
.
This is proved in [4] only in the case
.
A simple localisation argument then
completes the proof.

David J. Rule
School of Mathematics and Maxwell Institute for Mathematical Sciences
The University of Edinburghm
James Clerk Maxwell Building
The King's Buildings,
Mayfield Road
Edinburgh, EH9 3JZ, UK
email: rule@uchicago.edu